Continuity of cost in Borkar control topology and implications on discrete space and time approximations for controlled diffusions under several criteria Article Swipe
Somnath Pradhan
,
Serdar Yüksel
·
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.1214/24-ejp1093
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.1214/24-ejp1093
We first show that the discounted cost, cost up to an exit time, and ergodic cost involving controlled non-degenerate diffusions are continuous on the space of stationary control policies when the policies are given a topology introduced by Borkar [V. S. Borkar, A topology for Markov controls, Applied Mathematics and Optimization 20 (1989), 55–62]. The same applies for finite horizon problems when the control policies are Markov and the topology is revised to include time also as a parameter. We then establish that finite action/piecewise constant stationary policies are dense in the space of stationary Markov policies under this topology and the same holds for continuous policies. Using the above mentioned continuity and denseness results we establish that finite action/piecewise constant policies approximate optimal stationary policies with arbitrary precision. This gives rise to the applicability of many numerical methods such as policy iteration and stochastic learning methods for discounted cost, cost up to an exit time, and ergodic cost optimal control problems in continuous-time. For the finite-horizon setup, we establish additionally near optimality of time-discretized policies by an analogous argument. We thus present a unified and concise approach for approximations directly applicable under several commonly adopted cost criteria.
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- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1214/24-ejp1093
- https://projecteuclid.org/journals/electronic-journal-of-probability/volume-29/issue-none/Continuity-of-cost-in-Borkar-control-topology-and-implications-on/10.1214/24-EJP1093.pdf
- OA Status
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- Cited By
- 4
- References
- 37
- Related Works
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- OpenAlex ID
- https://openalex.org/W4392051468
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https://openalex.org/W4392051468Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.1214/24-ejp1093Digital Object Identifier
- Title
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Continuity of cost in Borkar control topology and implications on discrete space and time approximations for controlled diffusions under several criteriaWork title
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articleOpenAlex work type
- Language
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enPrimary language
- Publication year
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2024Year of publication
- Publication date
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2024-01-01Full publication date if available
- Authors
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Somnath Pradhan, Serdar YükselList of authors in order
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https://doi.org/10.1214/24-ejp1093Publisher landing page
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https://projecteuclid.org/journals/electronic-journal-of-probability/volume-29/issue-none/Continuity-of-cost-in-Borkar-control-topology-and-implications-on/10.1214/24-EJP1093.pdfDirect link to full text PDF
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YesWhether a free full text is available
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goldOpen access status per OpenAlex
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https://projecteuclid.org/journals/electronic-journal-of-probability/volume-29/issue-none/Continuity-of-cost-in-Borkar-control-topology-and-implications-on/10.1214/24-EJP1093.pdfDirect OA link when available
- Concepts
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Mathematics, Discrete time and continuous time, Topology (electrical circuits), Discrete space, Control (management), Approximations of π, Space (punctuation), Applied mathematics, Mathematical analysis, Combinatorics, Statistics, Computer science, Operating system, Artificial intelligenceTop concepts (fields/topics) attached by OpenAlex
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4Total citation count in OpenAlex
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2025: 1, 2024: 3Per-year citation counts (last 5 years)
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.first | 1 |
| abstract_inverted_index.given | 33 |
| abstract_inverted_index.gives | 130 |
| abstract_inverted_index.holds | 103 |
| abstract_inverted_index.space | 24, 92 |
| abstract_inverted_index.time, | 12, 155 |
| abstract_inverted_index.under | 97, 192 |
| abstract_inverted_index.Borkar | 38 |
| abstract_inverted_index.Markov | 45, 66, 95 |
| abstract_inverted_index.finite | 58, 83, 118 |
| abstract_inverted_index.policy | 141 |
| abstract_inverted_index.setup, | 167 |
| abstract_inverted_index.(1989), | 52 |
| abstract_inverted_index.Applied | 47 |
| abstract_inverted_index.Borkar, | 41 |
| abstract_inverted_index.adopted | 195 |
| abstract_inverted_index.applies | 56 |
| abstract_inverted_index.concise | 186 |
| abstract_inverted_index.control | 27, 63, 160 |
| abstract_inverted_index.ergodic | 14, 157 |
| abstract_inverted_index.horizon | 59 |
| abstract_inverted_index.include | 73 |
| abstract_inverted_index.methods | 138, 146 |
| abstract_inverted_index.optimal | 123, 159 |
| abstract_inverted_index.present | 182 |
| abstract_inverted_index.results | 114 |
| abstract_inverted_index.revised | 71 |
| abstract_inverted_index.several | 193 |
| abstract_inverted_index.unified | 184 |
| abstract_inverted_index.approach | 187 |
| abstract_inverted_index.commonly | 194 |
| abstract_inverted_index.constant | 85, 120 |
| abstract_inverted_index.directly | 190 |
| abstract_inverted_index.learning | 145 |
| abstract_inverted_index.policies | 28, 31, 64, 87, 96, 121, 125, 175 |
| abstract_inverted_index.problems | 60, 161 |
| abstract_inverted_index.topology | 35, 43, 69, 99 |
| abstract_inverted_index.55–62]. | 53 |
| abstract_inverted_index.analogous | 178 |
| abstract_inverted_index.arbitrary | 127 |
| abstract_inverted_index.argument. | 179 |
| abstract_inverted_index.controls, | 46 |
| abstract_inverted_index.criteria. | 197 |
| abstract_inverted_index.denseness | 113 |
| abstract_inverted_index.establish | 81, 116, 169 |
| abstract_inverted_index.involving | 16 |
| abstract_inverted_index.iteration | 142 |
| abstract_inverted_index.mentioned | 110 |
| abstract_inverted_index.numerical | 137 |
| abstract_inverted_index.policies. | 106 |
| abstract_inverted_index.applicable | 191 |
| abstract_inverted_index.continuity | 111 |
| abstract_inverted_index.continuous | 21, 105 |
| abstract_inverted_index.controlled | 17 |
| abstract_inverted_index.diffusions | 19 |
| abstract_inverted_index.discounted | 5, 148 |
| abstract_inverted_index.introduced | 36 |
| abstract_inverted_index.optimality | 172 |
| abstract_inverted_index.parameter. | 78 |
| abstract_inverted_index.precision. | 128 |
| abstract_inverted_index.stationary | 26, 86, 94, 124 |
| abstract_inverted_index.stochastic | 144 |
| abstract_inverted_index.Mathematics | 48 |
| abstract_inverted_index.approximate | 122 |
| abstract_inverted_index.Optimization | 50 |
| abstract_inverted_index.additionally | 170 |
| abstract_inverted_index.applicability | 134 |
| abstract_inverted_index.approximations | 189 |
| abstract_inverted_index.finite-horizon | 166 |
| abstract_inverted_index.non-degenerate | 18 |
| abstract_inverted_index.action/piecewise | 84, 119 |
| abstract_inverted_index.continuous-time. | 163 |
| abstract_inverted_index.time-discretized | 174 |
| cited_by_percentile_year.max | 97 |
| cited_by_percentile_year.min | 91 |
| countries_distinct_count | 1 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile.value | 0.86234069 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | True |